What condition may cause numbness coldness and tingling of the left arm?

Arm tingling. Various neuropathy syndromes and / or various radiculopathy syndromes may cause the symptoms. A neurological evaluation is a good idea. I would also recommend a complete collagen/vascular blood work up with rheumatological testing.
Numb. We worry about "thoracic outlet syndrome" but the person to best help is an orthopedic doctor. You might need an EMG by a neurologist as well. Start with the orthopedic doctor first. If you are displease with their plans, seek a neurologist.